## The Mathematics Portal

**Mathematics** is the study of numbers, quantity, space, pattern, structure, and change. Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.

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Leonhard Euler Image credit: Emanuel Handmann |

**Leonhard Euler** (pronounced oiler; IPA /ˈɔɪlər/) (April 15, 1707 Basel, Switzerland - September 18, 1783 St Petersburg, Russia) was a Swiss mathematician and physicist. He is considered to be the dominant mathematician of the 18th century and one of the greatest mathematicians of all time; he is certainly among the most prolific, with collected works filling over 70 volumes.

Euler developed many important concepts and proved numerous lasting theorems in diverse areas of mathematics, from calculus to number theory to topology. In the course of this work, he introduced many of modern mathematical terminologies, defining the concept of a *function*, and its notation, such as *sin*, *cos*, and *tan* for the trigonometric functions.

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The **knight's tour** is a mathematical chess problem in which the piece called the knight is to visit each square on an otherwise empty chess board exactly once, using only legal moves. It is a special case of the more general Hamiltonian path problem in graph theory. (A closely related non-Hamiltonian problem is that of the longest uncrossed knight's path.) The tour is called *closed* if the knight ends on a square from which it may legally move to its starting square (thereby forming an endless cycle), and *open* if not. The tour shown in this animation is open (see also a static image of the completed tour). On a standard 8 × 8 board there are 26,534,728,821,064 possible closed tours and 39,183,656,341,959,808 open tours (counting separately any tours that are equivalent by rotation, reflection, or reversing the direction of travel). Although the earliest known solutions to the knight's tour problem date back to the 9th century CE, the first general procedure for completing the knight's tour was Warnsdorff's rule, first described in 1823. The knight's tour was one of many chess puzzles solved by The Turk, a fake chess-playing machine exhibited as an automaton from 1770 to 1854, and exposed in the early 1820s as an elaborate hoax. True chess-playing automatons (i.e., computer programs) appeared in the 1950s, and by 1988 had become sufficiently advanced to win a match against a grandmaster; in 1997, Deep Blue famously became the first computer system to defeat a reigning world champion (Garry Kasparov) in a match under standard tournament time controls. Despite these advances, there is still debate as to whether chess will ever be "solved" as a computer problem (meaning an algorithm will be developed that can never lose a chess match). According to Zermelo's theorem, such an algorithm does exist.

## Did you know -

- ...that people are significantly slower to identify the parity of zero than other whole numbers, regardless of age, language spoken, or whether the symbol or word for zero is used?
- ...properties of Pascal's triangle have application in many fields of mathematics including combinatorics, algebra, calculus and geometry?
- ...work in artificial intelligence makes use of Swarm intelligence, which has foundations in the behavorial examples found in nature of ants, birds, bees, and fish among others?
- ...that statistical properties dictated by Benford's Law are used in auditing of financial accounts as one means of detecting fraud?
- ...that Modular arithmetic has application in at least ten different fields of study, including the arts, computer science, and chemistry in addition to mathematics?
- ... that according to Kawasaki's theorem, an origami crease pattern with one vertex may be folded flat if and only if the sum of every other angle between consecutive creases is 180º?

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The **Mathematics WikiProject** is the center for mathematics-related editing on Wikipedia. Join the discussion on the project's **talk page**.

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